Abstract
ABSTRACTAn outer-independent k-rainbow dominating function of a graph G is a function f from to the set of all subsets of such that both the following hold: (i) whenever v is a vertex with , and (ii) the set of all with is independent. The outer-independent k-rainbow domination number of G is the invariant , which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by an outer-independent k-rainbow dominating function. In this paper, we initiate the study of outer-independent k-rainbow domination. We first investigate the basic properties of the outer-independent k-rainbow domination and then we focus on the outer-independent 2-rainbow domination number and present sharp lower and upper bounds for it.
Highlights
We follow the notation and graph theory terminology in [1]
A leaf is a vertex of degree one, and a support vertex is a vertex adjacent to a leaf
If A ⊂ V(G), N(A) denotes the union of open neighbourhoods of all vertices of A. (If the graph G under consideration is not clear we write NG(u), and so on.) We denote by Pn and Cn the path and cycle on n vertices, respectively
Summary
We follow the notation and graph theory terminology in [1]. let G = (V(G), E(G)) be a finite simple graph. We denote the sets of all leaves and all support vertices of G by L(G) and S(G), respectively. By X we denote the induced subgraph of a graph G with vertex set X ⊆ V(G). A vertex cover of a graph G is a set of vertices that covers all the edges. Given a graph G, the minimum weight of a k-rainbow dominating function is called the k-rainbow domination number of G, which we denote by γrk(G). A k-rainbow dominating function f : V(G) → 2[k] is an outer-independent krainbow dominating function (an OIkRD-function) on G if the set {v ∈ V(G) | f (v) = ∅} is independent. The outerindependent k-rainbow domination number γokir(G) is the minimum weight of an OIkRD-function on G. We first investigate the basic properties of the outer-independent k-rainbow domination and we focus on the outer-independent 2-rainbow domination number and present sharp lower and upper bounds for it
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